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Abstract

The objective, in this work, is to study the alpha-norm, the existence, the continuity dependence in initial data, the regularity, and the compactness of solutions of mild solution for some semi-linear partial functional integrodifferential equations in abstract Banach space. Our main tools are the fractional power of linear operator theory and the operator resolvent theory. We suppose that the linear part has a resolvent operator in the sense of Grimmer. The nonlinear part is assumed to be continuous with respect to a fractional power of the linear part in the second variable. An application is provided to illustrate our results.

1. Introduction

We consider, in this manuscript, partial functional equations of retarded type with deviating arguments in terms of involving spatial partial derivatives in the following form [1]:

dutdt=−Aut+∫0tBt−susds+Ftutfort≥0,u0=φ∈Cα=C−r0DAα],E1

where −Ais the infinitesimal generator of an analytic semigroup Ttt≥0on a Banach space X. Btis a closed linear operator with domain DBt⊃DAtime-independent. For 0<α<1, Aαis the fractional power of Awhich will be precise in the sequel. The domain DAαis endowed with the norm ∥x∥α=∥Aαx∥called α−norm. Cαis the Banach space C−r0DAαof continuous functions from −r0to DAαendowed with the following norm:

∥ϕ∥α=sup−r≤θ≤0∥ϕθ∥αforϕ∈Cα.

F:R+×Cα→Xis a continuous function, and as usual, the history function ut∈Cαis defined by

utθ=ut+θforθ∈−r0.

As a model for this class, one may take the following Lotka-Volterra equation:

In the particular case where α=0, many results are obtained in the literature under various hypotheses concerning A, B, and F(see, for instance, [2, 3, 4, 5, 6] and the references therein). For example, in [7], Ezzinbi et al. investigated the existence and regularity of solutions of the following equation:

dutdt=−Aut+∫0tBt−susds+Ftutfort≥0,u0=φ∈C−r0X,E3

The authors obtained also the uniqueness and the representation of solutions via a variation of constant formula, and other properties of the resolvent operator were studied. In [8], Ezzinbi et al. studied a local existence and regularity of Eq. (3). To achieve their goal, the authors used the variation of constant formula, the theory of resolvent operator, and the principle contraction method. Ezzinbi et al. in [9] studied the local existence and global continuation for Eq. (3). Recall that the resolvent operator plays an important role in solving Eq. (3); in the weak and strict sense, it replaces the role of the c0semigroup theory. For more details in this topic, here are the papers of Chen and Grimmer [2], Hannsgen [10], Smart [11], Miller [12, 13], and Miller and Wheeler [14, 15]. In the case where the nonlinear part involves spatial derivative, the above obtained results become invalid. To overcome this difficulty, we shall restrict our problem in a Banach space Yα⊂X, to obtain our main results for Eq. (1).

Considering the case where B=0, Travis and Webb in [16] obtained results on the existence, stability, regularity, and compactness of Eq. (1). To achieve their goal, the authors assumed that −Ais the infinitesimal generator of a compact analytic semigroup and Fis only continuous with respect to a fractional power of A in the second variable. The present paper is motivated by the paper of Travis and Webb in [16].

The paper is organized as follows. In Section 2, we recall some fundamental properties of the resolvent operator and fractional powers of closed operators. The global existence, uniqueness, and continuous dependence with respect to the initial data are studied in Section 3. In Section 4, we study the local existence and bowing up phenomena. In Section 5 we prove, under some conditions, the regularity of the mild solutions. And finally, we illustrate our main results in Section 6 by examining an example.

2. Fractional power of closed operators and resolvent operator for integrodifferential equations

We shall write Yfor DAendowed with the graph norm xY=x+Ax,Yαfor DAαand LYαXwill denote the space of bounded linear operators from Yαto X, and for Y0=X, we write LXwith norm .LX. We also frequently use the Laplace transform of fwhich is denoted by f∗. If we assume that −Agenerates an analytic semigroup and, without loss of generality, that 0∈ϱA, then one can define the fractional power Aαfor 0<α<1, as a closed linear operator on its domain Yαwith its inverse A−αgiven by

3. Global existence, uniqueness, and continuous dependence with respect to the initial data

Definition 3.2. A continuous function u:0b→Yαis called a mild solution of Eq. (1) if

ut=Rtφ0+∫0tRt−sFsusdsfort∈0b,u0=φ∈Cα.E5

Now to obtain our first result, we take the following assumption.

(H1) There exists a constant LF>0such that

Ftφ1−F(tφ2)≤LFφ1−φ2αfort≥0andφ1,φ2∈Cα.

Theorem 3.3. Assume that (V1)–(V3) and (H1) hold. Then for φ∈Cα, Eq. (1) has a unique mild solution which is defined for all t≥0.

Proof. Let a>0. For φ∈Cα, we define the set ∧by

∧=y∈C0aYα:y0=φ0.

The set ∧is a closed subset of C0aYαwhere C0aYαis the space of continuous functions from 0ato Yαequipped with the uniform norm topology

∥y∥α=sup0≤t≤a∥yt∥αfory∈C0aYα.

For y∈∧, we introduce the extension y¯of yon −radefined by y¯t=ytfor t∈0aand y¯t=φtfor t∈−r0. We consider the operator Γdefined on ∧by

Γyt=Rtφ0+∫0tRt−sFsy¯sdsfort∈0a.

We claim that Γ∧⊂∧.In fact for y∈∧, we have Γy0=φ0, and by continuity of Fand Rtxfor x∈X, we deduce that Γy∈∧.In order to obtain our result, we apply the strict contraction principle. In fact, let u,v∈∧and t∈0a. Then

Then Γis a strict contraction on ∧, and it has a unique fixed point ywhich is the unique mild solution of Eq. (1) on 0a. To extend the solution of Eq. (1) in a2a, we show that the following equation has a unique mild solution:

Theorem 4.2. Suppose that (V1)–(V3), (H0), and (H2) hold. Moreover, assume that Fdefined from J×Ωinto Xis continuous where J×Ωis an open set in R+×Cα. Then for each φ∈Ω, Eq. (1) has at least one mild solution which is defined on some interval 0b.

Proof. Let φ∈Ω. For any real ζ∈Jand p>0,we define the following sets:

Iζ=t:0≤t≤ζandHp=ϕ∈Cα:ϕα≤p.

For ϕ∈Hp, we choose ζand psuch that tϕ+φ∈Iζ×Hpand Hp⊆Ω.By continuity of F, there exists N1≥0such that Ftϕ+φ≤N1for tϕin Iζ×Hp. We consider φ¯∈C−rζYαas the function defined by φ¯t=Rtφ0for t∈Iζand φ¯0=φ. Suppose that p¯<pand choose 0<b<ζsuch that

Notice that finding a fixed point of Sin K0is equivalent to finding a mild solution of Eq. (1) in K0. Furthermore, Sis a mapping from K0to K0, since if η∈K0we have Sη0=0and

Sηtα≤∫0tAαRt−sF(sηs+φ¯s)ds.

Then

Sηtα≤NαN1∫0tdst−sα≤NαN1∫0bdssα<p¯

which implies that SK0⊂K0. We claim that Sηt:η∈K0}is compact in Yαfor fixed t∈−rb.In fact, let βbe such that 0<α≤β<1. The above estimate show that AβSηt:η∈K0is bounded in X. Since Aα−βis compact operator, we infer that Aα−βAβSηt:η∈K0is compact in X,hence Sηt:η∈K0is compact in Yα. Next, we show that Sηt:η∈K0is equicontinuous. The equicontinuity of Sηt:η∈K0at t=0follows from the above estimation of Sηt. Now let 0<t0<t≤bwith t0be fixed. Then we have

We obtain the same results by taking t0be fixed with 0<t<t0≤b.Then we claim that limt→t0Sηt−Sηt0α=0uniformly in η∈K0which means that Sηt:η∈K0is equicontinuous. Then by Ascoli-Arzela theorem, Sη:η∈K0is relatively compact in K0. Finally, we prove that Sis continuous. Since Fis continuous, given ε>0, there exists δ>0, such that

sup0≤s≤bηs−η̂sα<δimpliesthatFsηs+φ¯s−F(sη̂s+φ¯s)<ε.

Then for 0≤t≤b, we have

Sηt−Sη̂tα≤Nα∫0t1t−sαFsηs+φ¯s−F(sη̂s+φ¯s)ds≤Nαε∫0tdssα.

This yields the continuity of S, and using Schauder’s fixed point theorem, we deduce that Shas a fixed point. Then the proof of the theorem is complete.

The following result gives the blowing up phenomena of the mild solution in finite times.

Theorem 4.3. Assume that (V1)–(V3), (H0), and (H2) hold and Fis a continuous and bounded mapping. Then for each φ∈Cα, Eq. (1) has a mild solution u.φon a maximal interval of existence −rbφ. Moreover if bφ<∞, then lim¯t→bφ−utφα=+∞.

Proof. Let u.φbe the mild solution of Eq. (1) defined on 0b. Similar arguments used in the local existence result can be used for the existence of b1>band a function u.ubdefined from bb1to Yαsatisfying

utub.φ=Rtubφ+∫btRt−sFsusdsfort∈bb1.

By a similar proceeding, we show that the mild solution u.φcan be extended to a maximal interval of existence −rbφ. Assume that bφ<+∞and lim¯t→bφ−utφα<+∞. There exists N2>0such that Fsus≤N2,for s∈0bφ. We claim that u.φis uniformly continuous. In fact, let 0<h≤t≤t+h<bφ. Then

ut+h−ut=Rt+h−Rtφ0+∫0tRt+h−s−Rt−sFsusds+∫tt+hRt+h−sFsusds.

By continuity of AαRt, we claim that AαRt+h−Rtφ0is uniformly continuous on each compact set. Moreover, Theorem 4.1 implies that AαRt+h−s−Rt−sFsus→0uniformly in twhen h→0.In fact we have

∫0tRt+h−s−Rt−sFsusαds≤∫0tRt+h−s−RhRt−sFsusαds+Rh−IAα∫0tRt−sF(sus)ds

Then using Theorem 4.1, we obtain that

∫0tRt+h−s−Rt−sFsusαds≤bφN2M∫0hdssα+Rh−IAα∫0tRt−sF(sus)ds.

We claim that the set {Aα∫0tRt−sFsusds:t∈[0,bφ)}is relatively compact. In fact, let tnn≥0be a sequence of 0bφ. Then there exist a subsequence tnkkand a real number t0such that tnk→t0. Using the dominated convergence theorem, we deduce that

∫0tnkAαRtnk−sFsusds→∫0t0AαRt0−sFsusds.

This implies that {Aα∫0tRt−sFsusds:t∈[0,bφ)}is relatively compact. Now using Banach-Steinhaus’ theorem, we deduce that

Rh−IAα∫0tRt−sFsusds→0

uniformly when h→0with respect to t∈0bφ. Moreover we have

∥∫tt+hRt+h−sFsusds∥α≤N2Nα∫0hdssα.

Consequently ut+h−utα→0ash→0uniformly in t∈0bφ. If h≤0, that is, for t≤t0, we have

one can show similar results by using the same reasoning. This implies that u.φis uniformly continuous. Therefore limt→bφ−utφexistsinYα.And consequently, u.φcan be extended to bφwhich contradicts the maximality of 0bφ.

The next result gives the global existence of the mild solutions under weak conditions of F. To achieve our goal, we introduce a following necessary result which is a consequence of Lemma 7.1.1 given in ([21], p. 197, Exo 4).

Then using Lemma 4.4, we deduce that u.φis bounded in 0bφ. Then we obtain that lim¯t→bφ−utφα<∞,which contradicts our hypothesis. Then the mild solution is global.

We focus now to the compactness of the flow defined by the mild solutions.

Theorem 4.6. Assume that (V1)–(V3) and (H0)–(H2) hold. Then the flow Utdefined from Cαto Cαby Utφ=ut.φis compact for t>r, where ut.φdenotes the mild solution starting from φ.

Proof. We use Ascoli-Arzela’s theorem. Let E=φγ:γ∈Γbe a bounded subset of Cαand let t>rbe fixed, but arbitrary. We will prove that UtE¯is compact. It follows from (H1) and inequality Eq. (7) that there exists N5such that

F(tutφγ)≤N2utφγ)+∥Ft0∥=N5forγ∈Γ.

For each γ∈Γ,we define fγ∈Cαby fγ=ut.φγ. We show now that for fixed θ∈−r0,the set fγθ:γ∈Γis precompact in Yα. For any γ∈Γ, we have

fγθ=Rt+θφγ0+∫0t+θRt+θ−sFsus.φds.

As Rtis compact for t>0, we need only to prove that the set

∫0t+θRt+θ−sF(sus(.φγ))ds:γ∈Γ

is compact. Also we have

μRε∫0t+θ−εRt+θ−ε−sF(sus(.φγ))ds:γ∈Γ=0,

where μis the measure of non-compactness. Moreover, using Theorem 4.1, we have

Thus Aβ∫t+θ−εt+θRt+θ−sF(sus(.φγ))ds:γ∈Γis a bounded subset of X. The precompactness in Yαnow follows from the compactness of A−β:X→Yα. Then the set UtEθ:−r≤θ≤0is precompacted in Yα. We prove that the family fγ:γ∈Γis equicontinuous. Let γin Γ,0<ε<t−r,and −r≤θ̂≤θ≤0with θ̂be fixed and h=θ−θ̂. Then

Using the compactness of the set Aα∫0t+θRt+θ−sF(sus(.φγ))ds:γ∈Γand the continuity of t→Rtxfor x∈X, the right side of the above inequality can be made sufficiently small for h>0small enough. Then we conclude that fγ:γ∈Γis equicontinuous. Consequently, by Ascoli-Arzela’s theorem, we conclude that the set Utφ:φ∈Eis compact, which means that the operator Utis compact for t>r.

5. Regularity of the mild solutions

We define the set Cα1by Cα1=C1−r0Yαas the set of continuously differentiable functions from −r0to Yα. We assume the following hypothesis.

(H3) Fis continuously differentiable, and the partial derivatives DtFand DφFare locally Lipschitz in the classical sense with respect to the second argument.

The set H=usws:s∈0ais compact in Cα. Since the partial derivatives of Fare locally Lipschitz with respect to the second argument, it is well-known that they are globally Lipschitz on H. Then we deduce that

ut−wtα≤Nαha∫0t1t−sαus−wsαds≤Nαha∫0t1t−sαsup0≤τ≤auτ−wταds,

where ha=LFNα+aNαLipDtF+aNαLipDφF,with LipDφFand LipDtFthe Lipschitz constant of DφFand DtF, respectively, which implies that

u−wα≤Nαha∫0adssαu−wα.

If we choose asuch that

Nαha∫0adssα<1,

then u=win 0a. Now we will prove that u=win 0+∞.Assume that there exists t0>0such that ut0≠wt0. Let t1=inft>0:∥ut−wt∥>0.By continuity, one has ut=wtfor t≤t1, and there exists ε>0such that ∥ut−wt∥>0for t∈t1t1+ε. Then it follows that for t∈t1t1+ε,

The −Ais a closed operator and generates an analytic compact semigroup Ttt≥0on X. Thus, there exists δin 0π/2and M≥0such that Λ=λ∈C:argλ<π2+δ∪0is contained in ρ−A, the resolvent set of −A, and Rλ−A<M/λfor λ∈Λ. The operator Btis closed and for x∈Y, Btx≤htxY. The operator Ahas a discrete spectrum, the eigenvalues are n2, and the corresponding normalized eigenvectors are enx=2πsinnx,n=1,2,⋯. Moreover the following formula holds:

Since λg1−1λλg1−1λI+A−1is bounded because g1−1λ∈Λ, then Aρλxhas the growth properties of g1−1λwhich tends to 1 if ∣λ∣goes to infinity. Then we deduce that Aρλ∈LX. Moreover, it is analytic from Λto LX. Now, for x∈Y, one has

Aρλx=g1−1λλg1−1λI+A−1AxandB∗λρλx=h∗λρλAx.

Then it follows that

Aρλx≤M/∣λ∣xYandB∗λρλ≤M/∣λ∣xY.

We deduce that Aρλ∈LYX, B∗λ=h∗λA∈LYX, and B∗λρλ∈LYX.Considering D=C0∞0π, we see that the conditions (V1)–(V3) and (H0) are verified. Hence the homogeneous linear equation of Eq. (18) has an analytic compact resolvent operator Rtt≥0. The function Fis continuous in the first variable from the fact that gis continuous in the first variable. Moreover from Lemma 6.1 and the continuity of g, we deduce that Fis continuous with respect to the second argument. This yields the continuity of Fin R+×C1/2. In addition, by assumption (H5) we deduce that

∥Ftφ1−Ftφ2∥≤rLf∥φ1−φ2∥C1/2.

Then Fis a continuous globally Lipschitz function with respect to the second argument. We obtain the following important result.

Proposition 6.2. Suppose that the assumptions (H4)–(H5) hold. Then Eq. (19) has a mild solution which is defined for t≥0.

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